In this paper we study different restrictions imposed over the set of permutations of size $n$, $S_n$, and for specific classes of restrictions study the cycle structure of corresponding permutations. More specifically, we prove that for any fixed positive integer $k$, the number of $k$-cycles of a uniformly chosen permutation $π\in S_n$ with the restriction "$π(i) \geq i-1$" for $i \in \{2, \ldots, n\}$ has a Normal asymptotic distribution. We further prove that this result translates into CLTs regarding multiplicities of fixed-size parts of a uniformly selected composition of $n$.